Copy time line markers to a new scene
Timeline markers are part of the scene data, you will only get the markers if you import the scene into another blend file.

Blender can have more than one scene in the same blend file, you could create a full copy of the current scene, remove the items you don't want and start your new animation. Similarly you could save a copy of your file and remove the un-needed items in the new file.

The other option is to use python to export/import the markers.

Run this in your current file to export the markers -

import bpy

markerfile = '/tmp/markertransfer.txt'

with open(markerfile, 'w') as mf:
    for m in bpy.context.scene.timeline_markers:
        mf.write('{},{}\n'.format(m.frame, m.name))


Then in your new file run this to get matching markers -

import bpy

markerfile = '/tmp/markertransfer.txt'

with open(markerfile, 'r') as mf:
    for l in mf:
        f,n = l.split(',')
        bpy.context.scene.timeline_markers.new(n.strip(), frame=int(f))